Controllers for regulated power inverters, AC/DC, and DC/DC converters

ABSTRACT

The present invention relates to methods and corresponding apparatus for regulation, control, and management of DC-to-AC, AC-to-DC, and/or DC-to-DC switching power conversion.

CROSS REFERENCES TO RELATED APPLICATIONS

This application is a continuation-in-part of U.S. patent application Ser. No. 15/803,471, filed 3 Nov. 2017, which is a continuation-in-part of U.S. patent application Ser. No. 15/646,692, filed 11 Jul. 2017 (now U.S. Pat. No. 9,843,271). This application also claims the benefit of the U.S. provisional patent applications 62/362,764 filed on 15 Jul. 2016, and 62/510,990 filed on 25 May 2017, all of which are incorporated in their entirety in the present application.

STATEMENT REGARDING FEDERALLY SPONSORED RESEARCH OR DEVELOPMENT

None.

COPYRIGHT NOTIFICATION

Portions of this patent application contain materials that are subject to copyright protection. The copyright owner has no objection to the facsimile reproduction by anyone of the patent document or the patent disclosure, as it appears in the Patent and Trademark Office patent file or records, but otherwise reserves all copyright rights whatsoever.

TECHNICAL FIELD

The present invention relates to methods and corresponding apparatus for regulated and efficient DC-to-AC conversion with high power quality, and to methods and corresponding apparatus for regulation and control of said DC-to-AC conversion. The invention further relates to methods and corresponding apparatus for regulation and control of AC-to-DC and/or DC-to-DC conversion.

BACKGROUND

A power inverter, or an inverter, or a DC-to-AC converter, is an electronic device or circuitry that changes direct current (DC) to alternating current (AC) (e.g. single- and/or 3-phase).

A power inverter would typically be a switching inverter that uses a switching device (solid-state or mechanical) to change DC to AC. There are may be many different power circuit topologies and control strategies used in inverter designs. Different design approaches address various issues that may be more or less important depending on the way that the inverter is intended to be used. For example, an inverter may use an H bridge that is built with four switches (solid-state or mechanical), that enables a voltage to be applied across a load in either direction.

Given a particular power circuit topology, an appropriate control strategy may need to be chosen in order to meet the desired inverter specifications, e.g. in terms of input and output voltages, AC and switching frequencies, output power and power density, power quality, efficiency, cost, and so on.

For example, one of the most obvious approaches to increase power density would be to increase the frequency that the inverter operates at (“switching frequency”). This may be done by employing wide-bandgap (WBG) semiconductors that use materials such as gallium nitride (GaN) or silicon carbide (SiC) and can function at higher power loads and frequencies, allowing for smaller, more energy-efficient devices.

A typical way to control a switching inverter is by mathematically defining voltage and/or current relations at different switching states to produce the “desired” output, obtaining the appropriate digitally sampled voltage and/or current values, then using numerical signal processing tools to implement an appropriate algorithm to control the switches [1]. However, such a straightforward approach often calls for performance compromises based on the ability to accurately define the voltage and current relations for various topological stages and load conditions (e.g. nonlinear loads), and to account for nonidealities and time variances of the components (since the values of, e.g., resistances, capacitances, inductances, switches' behavior, etc., may change in time due to the temperature changes, mechanical stress/vibrations, aging, etc.). In addition, a control processor for the inverter must meet a number of real-time processing challenges, especially at high switching frequencies, in order to effectively execute the algorithms required for efficient DC/AC conversion and circuit protection. Various design and implementation compromises are often made in order to overcome these challenges, and those can negatively impact the complexity, reliability, cost, and performance of the inverter.

Thus there is a need in a simple analog (i.e. continuous and real-time) controller that would allow us to bypass the detailed analysis of various topological stages during a switching cycle altogether, thus avoiding the pitfalls and limitations of straightforward digital techniques.

Further, there is a need in such a simple controller that (i) does not require any current sensors, or additional start-up and management means, (ii) provides robust, high quality (e.g. low voltage and current harmonic distortions), and well regulated AC outputs for a wide range of power factor loads, including highly nonlinear loads, and also (iii) offers multiple ways to optimize the cost-size-weight-performance tradespace.

SUMMARY

The present invention overcomes the limitations of the prior art by introducing an Inductor Current Mapping, or ICM controller. The ICM controller offers various overall advantages over respective digital controllers, and may provide multiple ways to optimize the cost-size-weight-performance tradespace.

In the detailed description that follows, we first introduce an idealized concept of an analog ICM controller for an H-bridge power inverter. We show how the “actual” voltage-current relations in the filter inductors may be directly “mapped,” by the constraints imposed by two Schmitt triggers, to the voltage relations among the inputs and the output of the controller's analog integrator, providing the desired inverter output that is effectively proportional to the reference voltage. We then discuss the basic operation and properties of this inverter and its associated ICM controller, using particular implementations for illustration. Further, we describe a simple controller modification that may improve the transient responses by utilizing a feedback signal proportional to the load current. Next, we discuss extensions of the ICM concept to other hard- or soft-switching power inverters (e.g. for 3-phase inverters), and for AC/DC and DC/DC converter topologies, and the use of ICM controllers in DC/DC converters for voltage, current, and power regulation.

Further scope and the applicability of the invention will be clarified through the detailed description given hereinafter. It should be understood, however, that the specific examples, while indicating preferred embodiments of the invention, are presented for illustration only. Various changes and modifications within the spirit and scope of the invention should become apparent to those skilled in the art from this detailed description. Furthermore, all the mathematical expressions, diagrams, and the examples of hardware implementations are used only as a descriptive language to convey the inventive ideas clearly, and are not limitative of the claimed invention.

BRIEF DESCRIPTION OF FIGURES

FIG. 1. Basic diagram of an asynchronous buck inverter of the present invention and its associated controller.

FIG. 2. Alternative grounding configuration of the inverter and its associated controller shown in FIG. 1.

FIG. 3. Illustrative steady-state voltage and current waveforms for the inverter and its associated controller shown in FIGS. 1 and 2. Low FCS frequency (6 kHz) is used for better waveform visibility.

FIG. 4. Illustrative transient voltage and current waveforms for the inverter and its associated controller shown in FIGS. 1 and 2. Low FCS frequency (6 kHz) is used for better waveform visibility.

FIG. 5. Illustrative schematic of a controller circuit implementation.

FIG. 6. Illustrative schematic of a particular implementation of the inverter and its associated controller shown in FIG. 1.

FIG. 7. Simulated MOSFET and total power losses, efficiency, and total harmonic distortions as functions of the output power for the particular inverter implementation shown in FIG. 6 (48 kHz FCS, resistive load).

FIG. 8. Illustrative steady-state voltage and current waveforms for a lagging PF=0.5 load (66.2 mH inductor in series with 14.4Ω resistor), for the inverter and its associated controller shown in FIG. 6 (48 kHz FCS).

FIG. 9. Illustrative transient voltage and current waveforms for a lagging PF load (50 muH inductor in series with 10.8Ω or 84.4Ω resistors), for the inverter and its associated controller shown in FIG. 6 (48 kHz FCS).

FIG. 10. Illustrative steady-state voltage and current waveforms for a leading PF=0.5 load (106 μF capacitor in series with 14.4Ω resistor), for the inverter and its associated controller shown in FIG. 6 (48 kHz FCS).

FIG. 11. Illustrative transient voltage and current waveforms for a leading PF load (70.7 μF capacitor in series with 21.6Ω or 1 MΩ resistors), for the inverter and its associated controller shown in FIG. 6 (48 kHz FCS).

FIG. 12. Illustrative schematic of a particular implementation of a free-running version of the inverter and its associated controller shown in FIG. 1.

FIG. 13. Power spectra of the common and differential mode output voltages for the particular FCS-based (FIG. 6) and free-running (FIG. 12) implementations. The vertical dashed lines are at 150 kHz.

FIG. 14. An H-bridge inverter with a free-running ICM controller.

FIG. 15. Example of full-range transient waveforms for ohmic load.

FIG. 16. Transient waveforms for nonlinear (full-wave diode rectifier) load (2 kW).

FIG. 17. Transient waveforms for changes in V_(ref) (“instantaneous” synchronization with reference).

FIG. 18. Illustrative steady state waveforms at full (2 kW) resistive load.

FIG. 19. Illustrative steady state waveforms at 10% (200 W) resistive load.

FIG. 20. Full-load (2 kW) waveforms on 2.8 ms to 3 ms interval.

FIG. 21. 10%-load (200 W) waveforms on 2.8 ms to 3 ms interval.

FIG. 22. PSD of inductor current at full and 10% loads.

FIG. 23. Efficiency and THD as functions of output power.

FIG. 24. Illustrative startup voltages and currents.

FIG. 25. Inverter and ICM controller with load current feedback.

FIG. 26. Transient responses for leading PF=0.5 load without (left) and with (right) current feedback.

FIG. 27. Buck-boost DC/DC converter with ICM controller.

FIG. 28. Example of transient waveforms for the converter shown in FIG. 27 configured for voltage, current, and power regulation.

FIG. 29. Startup voltages and currents for the buck-boost DC/DC converter with ICM controller.

FIG. 30. Illustrative example of an ICM-controlled 3-phase inverter.

FIG. 31. Example of full-range transient waveforms for the ICM-controlled 3-phase inverter shown in FIG. 30.

FIG. 32. Startup voltages and currents when the 3 kW ICM-controlled 3-phase inverter shown in FIG. 30 is connected to a low (p.f.=0.1) lagging power factor load in Δ configuration, with the apparent power of about 3 kW.

FIG. 33. Example of a single-phase H-bridge ICM-based AC/DC converter with high power factor and low harmonic distortions.

FIG. 34. Illustration of the behavior of a particular implementation of a 2 kW ICM-controlled single-phase AC/DC converter shown in FIG. 33.

FIG. 35. Illustration of variations in ICM controller topologies.

FIG. 36. Example of a 3-phase ICM-based AC/DC converter with high power factor and low harmonic distortions.

FIG. 37. Illustration of the behavior of a particular implementation of a 6 kW ICM-controlled 3-phase AC/DC converter shown in FIG. 36.

FIG. 38. PSDs of line currents and V_(CM)-V_(N) for a particular implementation of a 6 kW ICM-controlled 3-phase AC/DC converter shown in FIG. 36.

FIG. 39. Example of ICM-based control of a 3-phase AC/DC converter that is effectively equivalent to that shown in FIG. 36.

FIG. 40. Illustrative example of using the 3-phase AC/DC converters shown in FIG. 36 and FIG. 39 for 3-phase DC/AC conversion.

FIG. 41. Illustration of transient output power, voltages and currents, and the inductor currents, for a particular implementation of a 6 kW ICM-controlled 3-phase inverter shown in FIG. 40, in response to full-range (and independent of each other) step changes in the line-to-line resistive loads.

FIG. 42. Analog and digital ICM implementations.

FIG. 43. Illustration of effective equivalence of analog and digital ICM implementations.

FIG. 44. Illustration of effective equivalence of analog and digital ICM implementations for oversampled 8-bit digital representations of analog voltages (without lowpass filtering in the digital implementation).

FIG. 45. Principal schematic of a 3-phase 6-switch rectifier that incorporates the power stage, LC EMI filtering network (components within the dashed-line boundary), and low-power analog circuitry that provides both the control of the power stage and non-dissipative resonance damping of the EMI filters.

FIG. 46. Transient responses under mains voltage imbalance for a particular implementation of an ICM-based 3-phase rectifier shown in FIG. 45, for 115 VAC to 400 VDC conversion with ƒ_(AC)=500 Hz.

FIG. 47. Efficiency/losses breakdown, PF, and THD for a particular implementation of an ICM-based 3-phase rectifier shown in FIG. 45, for 115 VAC to 400 VDC conversion with ƒ_(AC)=500 Hz.

FIG. 48. Steady-state line currents and CM/DM voltages at 10 kW output for a particular implementation of an ICM-based 3-phase rectifier shown in FIG. 45, for 115 VAC to 400 VDC conversion with ƒ_(AC)=500 Hz.

FIG. 49. Steady-state switching voltages (at 10 kW output) for a particular implementation of an ICM-based 3-phase rectifier shown in FIG. 45, for 115 VAC to 400 VDC conversion with ƒ_(AC)=500 Hz.

FIG. 50. Three-level ICM control of an H-bridge inverter.

FIG. 51. Illustration of performance of a 3-level ICM-controlled H-bridge inverter of FIG. 50 in comparison with a respective 2-level inverter with the same power stage components.

FIG. 52. Grid setting/following 3-level ICM-controlled H-bridge converter used in the simulations of Section 11.

FIG. 53. Paralleled PV/battery ICM-based H-bridge converters powering nonlinear load.

FIG. 54. DC voltages V_(DC) and V′_(DC), the DC powers (P_(DC)=I_(DC)V_(DC) and P′_(DC)=I′_(DC)V_(DC)), the output (load) voltage V_(AC) and current I_(load), and the output voltage of the rectifier V_(rec) for the setup of FIG. 53.

ABBREVIATIONS

AC: alternating (current or voltage); A/D: Analog-to-Digital; ADC: Analog-to-Digital Converter (or Conversion);

BCM: Boundary Conduction Mode; BOM: Bill Of Materials;

CCM: Continuous Conduction Mode; CM: Common Mode; COTS: Commercial Off-The-Shelf;

DC: direct (current or voltage), or constant polarity (current or voltage); DCM: Discontinuous Conduction Mode; DCR: DC Resistance of an inductor; DM: Differential Mode; DSP: Digital Signal Processing/Processor;

EMC: electromagnetic compatibility; e.m.f.: electromotive force; EMI: electromagnetic interference; ESR: Equivalent Series Resistance;

FCS: Frequency Control Signal; FFT: Fast Fourier Transform; GaN: Gallium nitride;

ICM: Inductor Current Mapping; IGBT: Insulated-Gate Bipolar Transistor;

LC: inductor-capacitor;

MATLAB: MATrix LABoratory (numerical computing environment and fourth-generation programming language developed by MathWorks); MEA: More Electric Aircraft; MOS: Metal-Oxide-Semiconductor; MOSFET: Metal Oxide Semiconductor Field-Effect Transistor; MTBF: Mean Time Between Failures; NDL: Nonlinear Differential Limiter;

PF: Power Factor; PFC: Power Factor Correction; PFM: Pulse-Frequency Modulation; PoL: Point-of-Load; PSD: Power Spectral Density; PSM: Power Save Mode; PV: Photovoltaic/Photovoltaics; PWM: Pulse-Width Modulation/Modulator;

RFI: Radio Frequency Interference; RMS: Root Mean Square;

SCS: Switch Control Signal; SiC: Silicon carbide; SMPS: Switched-Mode Power Supply; SMVF: Switched-Mode Voltage Follower; SMVM: Switched-Mode Voltage Mirror; SNR: Signal to Noise Ratio; SCC: Switch Control Circuit;

THD: Total Harmonic Distortion;

UAV: Unmanned Aerial Vehicle; ULISR: Ultra Linear Isolated Switching Rectifier; ULSR(U): Ultra Linear Switching Rectifier (Unit);

VN: Virtual Neutral; VRM: Voltage Regulator Module; WBG: wide-bandgap;

ZVS: Zero Voltage Switching; ZVT: Zero Voltage Transition;

DETAILED DESCRIPTION 1 Illustrative Description of an Asynchronous Buck Inverter of the Present Invention and of its Principles of Operation

Let us first consider the simplified circuit diagram shown in FIG. 1.

This power inverter would be capable of converting the DC source voltage V_(in) into the AC output voltage V_(out) that is indicative of the AC reference voltage V_(ref).

In FIG. 1, the DC source voltage V_(in) is shown to be provided by a battery, and the resistor connected in series with the battery indicates the battery's internal resistance.

Capacitance of the capacitor connected in parallel to the battery would need to be sufficiently large (for example, of order 10 μF) to provide a relatively low impedance path for high-frequency current components. A significantly larger capacitance may be used (e.g., of order 1 mF, depending on the battery's internal resistance) to reduce the low frequency (e.g., twice the AC frequency) input current and voltage ripples.

In FIG. 1, the DC source voltage V_(in) is shown to be applied to the input of the H bridge comprising two pairs of switches (labeled “1” and “2”), and the output voltage of the H bridge is the switching voltage V*. For example, when the switches of the 2nd pair are “on” and the switches of the 1st pair are “off”, V* would be effectively equal to V_(in), and when the switches of the 1st pair are “on” and the switches of the 2nd pair are “off”, V* would be effectively equal to −V_(in).

The diodes explicitly shown in FIG. 1 as connected across the switches in the bridge would enable a non-zero current through the inductors when all switches in the bridge are “off”. All switches in the bridge being “off” may be viewed as asynchronous state of the inverter. During the asynchronous state, and depending on the magnitude and the direction of the current through the inductors, the switching voltage V* may have values between −V_(in) and V_(in).

One skilled in the art will recognize that, for example, if the switches are implemented using power MOSFETs, the diodes explicitly shown in FIG. 1 may be the MOSFET body diodes.

The switching voltage V* is further filtered with an LC filtering network to produce the output voltage V_(out). In FIG. 1, this network performs both differential mode (DM) filtering to produce the output differential voltage V_(out), and common mode (CM) filtering to reduce electromagnetic interference (EMI). The DM filtering bandwidth of the LC network should be sufficiently narrow to suppress the switching frequency and its harmonics, while remaining sufficiently larger than the AC frequency.

The DM inductance L in FIG. 1 may be provided by physical inductors, by leakage inductance of the CM choke, or by combination thereof.

The switches of the 1st and 2nd pairs in the bridge are turned “on” or “off” by the respective switch control signals (SCSs), labeled as Q₁ and Q₂, respectively, in FIG. 1. In the figure, it is implied that the switches are turned “on” by a high value of the respective SCS, and are turned “off” by the low value of the SCS.

In FIG. 1, the input to the integrator (with the integration time constant T) is a sum of the AC reference voltage V_(ref) and a voltage proportional to the switching voltage, μV*, and the output of the integrator contributes to the inputs of both inverting and non-inverting comparators. The comparators may also be characterized by sufficiently large hysteresis, e.g., be configured as Schmitt triggers.

When the comparators are configured as Schmitt triggers, the frequency control signal (FCS) V_(FCS) may be optional, as would be discussed further in the disclosure. Such a configuration of the inverter (without an FCS) would be a free-running configuration.

A (constant) positive threshold offset, or FCS offset, signal ΔV_(FCS) is added to the input of the non-inverting comparator in FIG. 1. This offset signal enables the SCSs Q₁ and Q₂ to simultaneously have low values, thus keeping all switches in the bridge “off”, and thus enabling an asynchronous state of the inverter.

A periodic FCS V_(FCS) may be added to the inputs of both inverting and non-inverting comparators to enable switching at typically constant (rather than variable) frequency, effectively equal to the FCS frequency.

A signal proportional to the output AC voltage,

${\mu\frac{\tau}{T}V_{out}},$ is added to the inputs of both inverting and non-inverting comparators for damping transient responses of the inverter caused by changes in the load (load current). The mechanism of such damping, and the choice of the time parameter τ, would be discussed further in the disclosure. One skilled in the art will recognize that, equivalently, a signal proportional to the time derivative of the output AC voltage, μτ{dot over (V)}_(out), may be added to the input of the integrator.

It may be important to point out that the output AC voltage V_(out) of the inverter of the present invention would be effectively independent of the DC source voltage V_(in), as long as (neglecting the voltage drops across the switches and the inductors) |V_(in)| is larger than |V_(out)|.

The grounding configuration of the inverter shown in FIG. 1 would be appropriate for the 240 V split phase configuration, similar to what would be found in North American households.

FIG. 2 shows the modification of the inverter appropriate for the 240 V-to-ground configuration, similar to what would be found in European and other households around the world.

One skilled in the art will recognize that the configuration shown in FIG. 2 would allow combining outputs of multiple such inverters into a multi-phase output.

FIG. 3 shows illustrative steady-state voltage and current waveforms for the inverter and its associated controller shown in FIGS. 1 and 2. Low FCS frequency (6 kHz) is used for better waveform visibility.

FIG. 4 shows illustrative transient voltage and current waveforms for the inverter and its associated controller shown in FIGS. 1 and 2. Low FCS frequency (6 kHz) is used for better waveform visibility.

In both FIG. 3 and FIG. 4, low FCS frequency (6 kHz) was used for better waveform visibility, and the switches were implemented using SiC power MOSFETs.

FIG. 5 provides an illustrative schematic of a controller circuit implementation.

One may see that this controller comprises an (inverting) integrator characterized by an integration time constant T, where the integrator input comprises a sum of (1) the signal proportional to the switching voltage, μV*, (2) the reference AC voltage V_(ref), and (3) the signal proportional to the time derivative of the output AC voltage, μτ{dot over (V)}_(out).

The controller further comprises two comparators outputting the SCSs Q₁ and Q₂.

A sum of the integrator output and the FCS V_(FCS) (which is a periodic triangle wave in this example) is supplied to the positive terminal of the comparator providing the output Q₁, and to the negative terminal of the comparator providing the output Q₂.

The reference thresholds for the comparators are provided by a resistive voltage divider as shown in the figure, and the threshold value supplied to the negative terminal of the comparator providing the output Q₂ is ΔV_(FCS) larger than the threshold value supplied to the positive terminal of the comparator providing the output Q₁.

To further illustrate the essential features of the asynchronous buck inverter of the present invention, and of its associated controller, let us consider a particular implementation example shown in FIG. 6.

Note that in this example the capacitance of the capacitor connected in parallel to the battery is only 10 μF. It would provide a relatively low impedance path for high-frequency current components. However, a significantly larger capacitance (e.g., of order 1 mF, depending on the battery's internal resistance) should be used to reduce the low frequency (e.g. twice the AC frequency) input current and voltage ripples.

FIG. 7 shows simulated MOSFET and total power losses, efficiency, and total harmonic distortions as functions of the output power for the particular inverter implementation shown in FIG. 6 (48 kHz FCS, resistive load).

FIG. 8 provides illustrative steady-state voltage and current waveforms for a lagging PF=0.5 load (66.2 mH inductor in series with 14.4Ω resistor), for the inverter and its associated controller shown in FIG. 6 (48 kHz FCS).

FIG. 9 provides illustrative transient voltage and current waveforms for a lagging PF load (50 mH inductor in series with 10.8Ω or 84.4Ω resistors), for the inverter and its associated controller shown in FIG. 6 (48 kHz FCS).

FIG. 10 provides illustrative steady-state voltage and current waveforms for a leading PF=0.5 load (106 μF capacitor in series with 14.4Ω resistor), for the inverter and its associated controller shown in FIG. 6 (48 kHz FCS).

FIG. 11 provides illustrative transient, voltage and current waveforms for a leading PF load (70.7 μF capacitor in series with 21.6Ω or 1 MΩ resistors), for the inverter and its associated controller shown in FIG. 6 (48 kHz FCS).

2 Free-Running (Variable Frequency) Configuration of an Inverter of the Present Invention

FIG. 12 shows an illustrative schematic of a particular implementation of a free-running version of the inverter and its associated controller shown in FIG. 1.

In a free-running configuration, the switching frequency varies with the output voltage, and its average value would depend on the input and output voltages and the load, and would be generally inversely proportional to the value of the hysteresis gap of the Schmitt triggers and the time constant of the integrator.

With the components and the component values shown in FIG. 12, the inverter typically (e.g., in a steady state) operates in a discontinuous conduction mode (DCM) for the output powers ≲2.2 kW, and the modulation type may be viewed as effectively a pulse-frequency modulation.

To illustrate the differences in switching behavior, FIG. 13 shows the power spectra of the common (CM) and differential mode (DM) output voltages for the particular FCS-based (FIG. 6) and free-running (FIG. 12) implementations. The vertical dashed lines are at 150 kHz.

3 Detailed Description of Basic Operation of an H-Bridge Inverter with a Free-Running ICM Controller

FIG. 14 shows a simplified schematic of an H-bridge inverter with a free-running ICM controller. As may be seen in the figure, the MOSFET switches in the bridge are turned on or off by high or low values, respectively, of the switch control signals (SCS) Q₁ and Q₂ provided by the controller, and the bridge outputs the switching voltage V*. The switching voltage is then filtered with a passive LC filtering network to produce the output voltage V_(out). The ICM controller effectively consists of (i) an integrator (with the time constant T) integrating a sum of three inputs, and (ii) two Schmitt triggers with the same hysteresis gap Δh, inverting (outputting Q₁) and non-inverting (outputting Q₂). A resistive divider ensures that the reference voltage of the non-inverting Schmitt trigger is slightly higher than that of the inverting one.

For the performance examples in this section that follow, the components and their nominal values are as follows:

The battery electromotive force (e.m.f.) is ε=450 V; the battery's internal resistance is 10Ω; the capacitance of the capacitor connected in parallel to the battery is 1,200 μF; the switches are the Cree C2M0025120D SiC MOSFETs; the diodes in parallel to the switches are the Rohm SCS220KGC SiC Schottky diodes; the OpAmps are LT1211; the comparators are LT1715; L=195 μH; C=4 μF; μ= 1/200; R=10 kΩ; τ=47 μs (τ/R=4.7 nF); T=22 μs (T/R=2.2 nF); V_(s)=5 V; R′=10 kΩ; δR′=510Ω; r=100 kΩ, and δr=10 kΩ.

Let us initially make a few sensible idealizations, including nearly ideal behavior of the controller circuit, insignificant voltage drops across the components of the bridge, negligible ohmic voltage drops across the reactive components, and approximately constant L and C. We would also assume a practical choice of T and Δh that would ensure sufficiently high switching frequencies for the switching ripples in the output voltage to be ignored, as they would be adequately filtered by the LC circuit. This would imply that we may consider an instantaneous value of V_(out) within a switching cycle to be effectively equal to the average value of V_(out) over this cycle. With such idealizations, and provided that |V_(ref)|<μ|V_(in)| at all times, the output voltage V_(out) may be expressed, in reference to FIG. 14, by the following equation:

$\begin{matrix} {{{\overset{\_}{V}}_{out} = {\left( {{- \frac{{\overset{\_}{V}}_{ref}}{\mu}} - {2L{\overset{\overset{.}{\_}}{I}}_{load}}} \right) - {\tau\;{\overset{\overset{.}{\_}}{V}}_{out}} - {2{LC}\;{\overset{\overset{¨}{\_}}{V}}_{out}}}},} & (1) \end{matrix}$ where I_(load) is the load current, the overdots denote time derivatives, μ is the nominal proportionality constant between the reference and the output voltages, and the overlines denote averaging over a time interval between a pair of rising or falling edges of either Q₁ or Q₂. Thus, according to equation (1), the output voltage of the inverter shown in FIG. 14 would be equal to the voltage −V _(ref)/μ−2Lİ _(load) filtered with a 2nd order lowpass filter with the undamped natural frequency ω_(n)=1/√{square root over (2LC)} and the quality factor Q=√{square root over (2LC)}/τ. The reference voltage V_(ref) in equation (1) may be an internal reference (e.g. for operating in an islanded mode), or an external reference (e.g. it may be proportional to the mains voltage for synchronization with the grid).

3.1 Derivation of Equation (1)

Let us show how equation (1) may be derived by mapping the voltage and current relations in the output LC filter to the voltage relations among the inputs and the output of the integrator in the ICM controller.

Indeed, for a continuous function ƒ(t), the time derivative of its average over a time interval ΔT may be expressed as

$\begin{matrix} {{{\overset{\overset{.}{\_}}{f}(t)} = {{\frac{d}{dt}\left\lbrack {\frac{1}{\Delta\; T}{\int_{t - {\Delta\; T}}^{t}{{dsf}(s)}}} \right\rbrack} = {\frac{1}{\Delta\; T}\left\lbrack {{f(t)} - {f\left( {t - {\Delta\; T}} \right)}} \right\rbrack}}},} & (2) \end{matrix}$ and it will be zero if ƒ(t)−ƒ(t−ΔT)=0.

Note that rising and falling edges of an output of a Schmitt trigger happen when its input crosses (effectively constant) respective thresholds. Thus relating the inputs and the output of the integrator in the ICM controller, differentiating both sides, then averaging between a pair of rising or falling edges of either Q₁ or Q₂, would lead to

$\begin{matrix} {{\overset{\_}{V}}^{*} = {{- \frac{{\overset{\_}{V}}_{ref}}{\mu}} - {\tau\;{{\overset{\overset{.}{\_}}{V}}_{out}.}}}} & (3) \end{matrix}$ On the other hand, from the voltage and current relations in the output LC filter, V*=V _(out)+2L İ _(load)+2LC {umlaut over (V)} _(out),  (4) and equating the right-hand sides of (3) and (4) would lead to equation (1).

3.2 Transient Responses

For convenience, we may define the “no-load ideal output voltage” V_(ideal) as

$\begin{matrix} {V_{ideal} = {{- \frac{V_{ref}}{\mu}} - {\tau\;{\overset{.}{V}}_{ideal}} - {2{LC}\;{{\overset{¨}{V}}_{ideal}.}}}} & (5) \end{matrix}$ Then equation (1) may be rewritten as

$\begin{matrix} {{\overset{\_}{v} = {{{- \frac{2\;\mu\; L}{V_{0}}}{\overset{\overset{.}{\_}}{I}}_{load}} - {\tau\;\overset{\overset{.}{\_}}{v}} - {2{LC}\;\overset{\overset{¨}{\_}}{v}}}},} & (6) \end{matrix}$ where ν is a nondimensionalized error voltage that may be defined as the difference between the actual and the ideal outputs in relation to the magnitude of the ideal output, for example, as

$\begin{matrix} {{v = \frac{V_{out} - V_{ideal}}{\max{V_{ideal}}}},} & (7) \end{matrix}$ where max |V_(ideal)| is the amplitude of the ideal output. For a sinusoidal reference V_(ref)=V₀ sin(2πƒ_(AC)t), and provided that Q≲1 and τƒ_(AC)<<<1, the no-load ideal output would be different from −V_(ref)/μ by only a relatively small time delay and a negligible change in the amplitude. Then ν may be expressed as ν=μ(V_(out)−V_(ideal))/V₀.

Note that the symbol ≲ may be read as “smaller than or similar to”, and the symbol <<< may be understood as “two orders or more smaller than”.

Thus, according to equation (6), the transients in the output voltage (in addition to the “normal” switching ripples at constant load) would be proportional to the time derivative of the load current filtered with a 2nd order lowpass filter with the undamped natural frequency ω_(n)=1/√{square root over (2LC)} and the quality factor Q=√{square root over (2LC)}/τ. Note that the magnitude of these transients would also be proportional to the output filter inductance L.

For example, for a step change at time t=t₀ in the conductance G of an ohmic load, from G=G₁ to G=G₂, equation (6) would become

$\begin{matrix} {{\overset{\_}{v} = {{{- \frac{2\mu\;{{LV}_{out}\left( t_{0} \right)}}{V_{0}}}\left( {G_{2} - G_{1}} \right)\overset{\_}{\delta\left( {t - t_{0}} \right)}} - {\left\{ {\tau + {2{L\left\lbrack {{G_{1}\overset{\_}{\theta\left( {t_{0} - t} \right)}} + {G_{2}\overset{\_}{\theta\left( {t - t_{0}} \right)}}} \right\rbrack}}} \right\}\overset{\overset{.}{\_}}{v}} - {2{LC}\;\overset{\overset{¨}{\_}}{v}}}},} & (8) \end{matrix}$ where θ(x) is the Heaviside unit step function [2] and δ(x) is the Dirac δ-function [3]. In equation (8), the first term is the impulse disturbance due to the step change in the load current, and V_(out)(t₀) is the output voltage at t=t₀. Note that δ(t−t₀) would be zero if to lies outside of the averaging interval ΔT, and would be equal to 1/ΔT otherwise.

Note that formally δ(t−t₀)=1/(2ΔT) when to is exactly at the beginning or at the end of the averaging interval.

FIG. 15 provides an example of illustrative transient voltage and current waveforms for a particular implementation of the inverter and its associated ICM controller shown in FIG. 14, in response to full-range step changes in a resistive load. Note that the switching interval, as well as the inductor current and its operation mode (e.g. continuous (CCM) or discontinuous (DCM)) change according to the output voltage values and the load conditions, and would also change according to the power factor of the load.

It is also instructive to illustrate the robustness and stability of this inverter for other highly nonlinear loads. (A nonlinear electrical load is a load where the wave shape of the steady-state current does not follow the wave shape of the applied voltage (i.e. impedance changes with the applied voltage and Ohm's law is not applicable)). FIG. 16 provides an example of voltage and current waveforms when the inverter with its associated ICM controller shown in FIG. 14 is connected to a full-wave diode rectifier powering a 2 kW load.

FIG. 17 provides an example of transient voltage and current waveforms, for the same inverter, in response to connecting and disconnecting the reference voltage. This shows the “instantaneous” synchronization with reference, e.g., illustrating how the presented inverter may quickly disconnect from and/or reconnect to the grid when used as a grid-tie inverter.

3.3 Switching Behavior and Efficiency Optimization

From the above mathematical description one may deduce that, for given inductor and capacitor values, and for given controller parameters (e.g., T and Δh), the total switching interval (i.e., the time interval between an adjacent pair of rising or falling edges of either Q₁ or Q₂), the duty cycle, and the “on” and “off” times, would all vary depending on the values and time variations of V_(in), V_(ref), V_(out), and the load current. Thus the switching behavior of the ICM controller may not be characterized in such simple terms as pulse-width or pulse-frequency modulation (PWM or PFM). This is illustrated in FIGS. 18 through 21, which show representative steady state waveforms at full (2 kW), and 10% (200 W) resistive loads.

In a steady state (i.e. for a constant load), the average value of the switching interval would be generally proportional to the product of the integrator time constant T and the hysteresis gap Δh of the Schmitt triggers (i.e. the values of r and δr). The particular value of the switching interval would also depend on the absolute value of the reference voltage |V_(ref)| (as ∝|V_(ref)|⁻¹), on the ratio |V_(ref)|/|V_(in)|, and on the load current [4, 5]. As a result, even in a steady state, for a sinusoidal AC reference the switching frequency would span a continuous range of values, as illustrated in FIG. 22, which shows the power spectral density (PSD) of the inductor current at full (2 kW) and 10% (200 W) resistive loads.

For given inverter components and their values, the power losses in various components would be different nonlinear functions of the load, and would also exhibit different nonlinear dependences on the integrator time constant T and the hysteresis gap Δh. Thus, given a particular choice of the MOSFET switches and their drivers, the magnetics, and other passive inverter components, by adjusting T and/or Δh one may achieve the best overall compromise among various component power losses (e.g., between the bridge and the inductor losses), while remaining within other constraints on the inverter specifications.

FIG. 23 provides an example of simulated efficiency and total harmonic distortion (THD) values as functions of the output power for a particular inverter implementation using commercial off-the-shelf (COTS) components (including SiC MOSFETs and diodes), with the specifications according to the technical requirements for the Little Box Challenge [6] outlined in [7]. In the efficiency simulations, high-fidelity models were used for the MOSFETs and diodes, and the inductor core and winding losses were taken into account. The dashed line in the upper panel of FIG. 23 plots the simulated efficiency with more conservative (doubled) estimated total losses, including the MOSFET and the inductor losses.

3.4 Startup Behavior

FIG. 24 illustrates the startup voltages (upper panel) and currents (lower panel) for a full (2 kW) resistive load connected to the output of the inverter. As one may see, as long as the controller is powered up, the battery may be connected to the input of the inverter (at 12.5 ms in the figure), and the output voltage would quickly converge to the desired output without excessive inrush currents and/or voltage transients. The only significant inrush current may be the initial current through the battery, charging the inverter's input capacitor during the time interval comparable with the product of the input capacitor and the battery's internal resistance.

3.5 Improving Transient Response by Introducing Feedback of the Load Current

One may infer from equation (1) that the term −2Lİ _(load) may be cancelled by adding −2 μLİ_(load) to the input of the integrator. However, the switching ripples in the load current would normally make such an approach impractical. Instead, one should add a voltage 2μLI_(load)/T directly to the inputs of the Schmitt triggers, as illustrated in FIG. 25.

Additionally, because of the propagation delays and other circuit nonidealities, fast step current transients may not be cancelled exactly. Instead, the current feedback would try to “counteract” an impulse disturbance in the output voltage due to a step change in the load current by a closely following pulse of opposite polarity, mainly reducing the frequency content of the transients that lies below the switching frequencies. This is illustrated in FIG. 26, which compares the transient responses for an ICM controller without (left) and with (right) current feedback, when a leading PF=0.5 load is connected to and disconnected from the output of the inverter with the frequency 1 kHz and 50% duty cycle. As indicated by the horizontal lines in the lower panels of the example of FIG. 26, the current feedback reduces the PSD of the transients at the load switching frequency (1 kHz) by approximately 12 dB.

4 Buck-Boost DC/DC Converter with ICM Controller

While above the ICM controller is disclosed in connection with a hard switching H-bridge power inverter, the ICM concept may be extended, with proper modifications, to other hard- or soft-switching power inverter and DC/DC converter topologies.

As an illustration, FIG. 27 provides an example of a buck-boost DC/DC converter with an ICM controller. When the feedback voltage V_(fb), is proportional to the output voltage, V_(fb)=−V_(out)/β, this converter would provide an output voltage regulation with the nominal steady-state output voltage βV_(ref). When the feedback voltage V_(fb), is proportional to the load current, V_(fb)=−

I*/β, this converter would provide an output current regulation with the nominal steady-state output current βV_(ref)/

. Note that in this example the load explicitly contains a (parallel) capacitance that may be comparable with, or larger than, the converter capacitance C, and thus the feedback voltage may contain very strong high-frequency components. Further, when the feedback voltage V_(fb) is proportional to the output power, V_(fb)=−V_(out)I*/I₀/β, this converter would provide an output power regulation with the nominal steady-state output power βV_(ref)I₀.

FIG. 28 provides an example of transient waveforms for a particular implementation of the converter shown in FIG. 27 configured for voltage, current, and power regulation.

For an isolated version, the converter inductor may be replaced by a flyback transformer, as indicated in the upper right corner of FIG. 27.

4.1 Startup Sequence for the Buck-Boost DC/DC Converter with ICM Controller

For a proper startup of the converter shown in FIG. 27, the initial value of the reference voltage V_(ref) should be zero. After both the controller circuit and the power stage are powered up, the reference voltage should be ramped up from zero to the desired value over some time interval. The duration of this time interval would depend on the converter specifications and the component values, and would typically be in the 1 ms to 100 ms range.

For a particular implementation of the converter, FIG. 29 illustrates the startup voltages and currents for a full resistive load connected to the output of the converter (upper panel), and for no load at the output (lower panel). In this illustration, the controller circuit is powered up first, with the reference voltage set to zero. At 5 ms, the battery is connected to the converter input. At 10 ms, the reference voltage starts ramping up from zero, reaching its desired value at 20 ms. As one may see, with such a startup sequence there would be no excessively high inrush currents through the converter inductor.

5 ICM Control of 3-Phase Inverters

FIG. 30 provides an example of an ICM-controlled 3-phase inverter. Here, the output line-to-line voltages V_(ab), V_(bc), and V_(ca) would be proportional to the respective reference voltages

_(ab),

_(bc), and

_(ca), where

_(ab)+

_(bc)+

_(ca)=0. Note that, in general, the reference voltages do not need to be sinusoidal signals.

As one should be able to see in FIG. 30, the switches in the bridge are controlled by three instances of an ICM controller disclosed herein, and each leg of the bridge is separately controlled by its respective ICM controller.

The switching voltages V*_(ab)V*_(bc), and V*_(ca) are the differences between the voltages at nodes a, b, and c: V*_(ab)=V*_(a)−V*_(b), V*_(bc)=V*_(b)−V*_(c), and V*_(ca)=V*_(c) 31 V*_(a).

This inverter is characterized by the advantages shared with the H-bridge inverter presented above: robust, high quality, and well-regulated AC output for a wide range of power factor loads, voltage-based control without the need for separate start-up management, the ability to power highly nonlinear loads, effectively instantaneous synchronization with the reference (allowing to quickly disconnect from and/or reconnect to the grid when used as a grid-tie inverter), and multiple ways to optimize efficiency and the cost-size-weight-performance tradespace.

For example, FIG. 31 illustrates transient output voltages and currents, and the inductor currents, for a particular implementation of a 3 kW ICM-controlled 3-phase inverter shown in FIG. 30, in response to full-range (and independent of each other) step changes in the line-to-line resistive loads.

Note that for 3-phase loads that are not significantly unbalanced, the value of the input capacitor may be significantly reduced, in comparison with the single-phase H-bridge inverter, without exceeding the limits on the input current ripples.

FIG. 32 illustrates the startup voltages and currents when the 3 kW ICM-controlled 3-phase inverter shown in FIG. 30 is connected to a low (p.f.=0.1) lagging power factor load in Δ configuration, with the apparent power of about 3 kW. Note that the transient current through the battery is bi-directional, going through the discharge-recharge cycles as needed, before the steady-state operation is achieved.

6 Bidirectionality of ICM-Controlled Inverters and AC/DC Converters

FIG. 33 provides an example of a single-phase H-bridge ICM-based AC/DC converter with high power factor and low harmonic distortions.

As before, let us make a few sensible idealizations, including nearly ideal behavior of the controller circuit, insignificant voltage drops across the components of the bridge, negligible ohmic voltage drops across the reactive components, and constant inductances L and the output capacitance C_(out). We also assume a practical choice of the integrator time constant T in the ICM controller circuit, and the hysteresis gap Δh of the Schmitt triggers, that ensures sufficiently high switching frequencies for the switching ripples in the output voltage to be ignored. This also implies that we can consider an instantaneous value of V_(AC) within a switching cycle to be effectively equal to the average value of V_(AC) over this cycle.

Note that rising and falling edges of an output of a Schmitt trigger happen when its input crosses (effectively constant) respective thresholds. Thus relating the inputs and the output of the integrator in the ICM controller in FIG. 33, differentiating both sides, then averaging between a pair of rising or falling edges of either Q₁ or Q₂, would lead to V*=V _(AC)−βτ {dot over (V)} _(AC)  (9) On the other hand, the line current I_(AC) in FIG. 33 may be related to the line and switching voltages V_(AC) and V* according to the following equation:

$\begin{matrix} {{{\overset{\_}{I}}_{A\; C} = {\frac{1}{2L}{\int{{dt}\left( {{\overset{\_}{V}}_{A\; C} - {\overset{\_}{V}}^{*}} \right)}}}},} & (10) \end{matrix}$ and substituting V* from equation (9) into (10) would lead to

$\begin{matrix} {{{\overset{\_}{I}}_{A\; C} = {{\beta\frac{\tau}{2L}{\overset{\_}{V}}_{A\; C}} + {const}}},} & (11) \end{matrix}$ where the constant of integration would be determined by the initial conditions and would decay to zero, due to the power dissipation in the converter's components and the load, for a steady-state solution.

Thus in a steady state

${{\overset{\_}{I}}_{A\; C} = {{\beta\;\frac{\tau}{2L}{\overset{\_}{V}}_{A\; C}} \propto {\overset{\_}{V}}_{A\; C}}},$ leading to AC/DC conversion with effectively unity power factor and low harmonic distortions. Note that high power factor in the ICM-based AC/DC converter shown in FIG. 33 is achieved without need for current sensing, and the PFC is entirely voltage-based.

The output parallel RC circuit forms a current filter which, with respect to the input current fed by a current source, acts as a 1st order lowpass filter with the time constant τ_(out)=R_(load)C_(out), and thus, for sufficiently large τ_(out) (e.g. an order of magnitude larger than ƒ_(AC) ⁻¹), the average value of the output voltage V_(out) would be proportional to the RMS of the line voltage and may be expressed as

$\begin{matrix} {{\left\langle V_{out} \right\rangle = {{\left( {{\eta\beta}\; R_{load}\frac{\tau}{2L}} \right)^{\frac{1}{2}}\left\langle V_{A\; C}^{2} \right\rangle^{\frac{1}{2}}} = {K\left\langle V_{A\; C}^{2} \right\rangle^{\frac{1}{2}}}}},} & (12) \end{matrix}$ where the angular brackets denote averaging over sufficiently large time interval (e.g. several AC cycles), η is the converter efficiency, and where

V_(AC) ²

^(1/2) is the RMS of the line voltage.

For example, for L=195 μH and τ=13.7 μs, 2L/τ≈28.5Ω. Thus for β=1 and 95% efficiency R_(load)=120Ω would result in K≈2.

For regulation of the output voltage V_(out), the load conductance may be obtained by sensing both the output voltage and the load current I_(load), and the coefficient β may be adjusted and/or maintained to be proportional to the ratio I_(load)/V_(out).

For example, from equation (12),

$\begin{matrix} {\beta = {{\frac{2K^{2}L}{\eta\tau}R_{load}^{- 1}} = {\frac{2K^{2}L}{\eta\tau}\frac{I_{load}}{V_{out}}}}} & (13) \end{matrix}$ would lead to the nominal AC/DC conversion ratio

V_(out)

/

V_(AC) ²

^(1/2)=K.

FIG. 34 illustrates the behavior of a particular implementation of a 2 kW ICM-controlled single-phase AC/DC converter shown in FIG. 33. In this example, the load conductance switches (effectively instantaneously) between zero and the full load, and the coefficient β is obtained according to equation (13). One should be able to see that, after the converter is powered up (after the AC source is connected at t=2.75 ms), the output voltage converges to the value given by equation (12), and the line current converges to the steady-state current given by equation (11) with const=0. The bottom panel in FIG. 34 shows the PSD of the steady-state current at full load, illustrating that the switching frequency would span a continuous range of values.

Further, additional output voltage regulation based on the difference between the desired nominal (“reference”) and the actual (

V_(out)

) output voltages may be added. For example, a term proportional to said difference may be added to the parameter 3.

7 Variations of ICM Controller Topologies and 3-Phase AC/DC and DC/AC Converters

One skilled in the art will recognize that an ICM controller allows mapping of the voltage relations among the inputs and the outputs of the integrators in the ICM controller into various desired voltage and current relations in a converter.

For example, as illustrated in FIG. 35(a), given a plurality of integrator inputs and a plurality of comparator (Schmitt trigger) inputs added, along with the integrator output, to the comparator (Schmitt trigger) input, these pluralities of inputs may be related by the following differential equation:

$\begin{matrix} {{{\frac{1}{T}{\sum\overset{\_}{\left( {{plurality}\mspace{14mu}{of}\mspace{14mu}{integrator}\mspace{14mu}{inputs}} \right)}}} = {\frac{d}{dt}{\sum\overset{\_}{\left( {{plurality}\mspace{14mu}{of}\mspace{14mu}{added}\mspace{14mu}{comparator}\mspace{14mu}{inputs}} \right)}}}},} & (14) \end{matrix}$ where the overlines denote averaging over a time interval between a pair of rising or falling edges of either first or second Schmitt trigger. Then (as illustrated, for example, in Sections 3.1 and 6) the desired voltage and current relations in a converter may be mapped (typically, through the relation to a switching voltage(s)) into the voltage relations in an ICM controller.

Note that, in accordance with equation (14), the ICM controllers shown in FIGS. 35(b) and 35(c) would be effectively equivalent.

In FIG. 35(b) the integrator input signal comprises a first integrator input component proportional to the switching voltage (μV*), a second integrator input component proportional to the AC source voltage (−μV_(AC)), and a third integrator input component proportional to a time derivative of said AC source voltage (βμτ{dot over (V)}_(AC)), and the Schmitt trigger input signal comprises a first Schmitt trigger input component proportional to said integrator output signal and an optional second Schmitt trigger input component proportional to a frequency control signal (V_(FCS)).

In FIG. 35(c) the integrator input signal comprises a first integrator input component proportional to the switching voltage (μV*) and a second integrator input component proportional to the AC source voltage (−μV_(AC)), and the Schmitt trigger input signal comprises a first Schmitt trigger input component proportional to said integrator output signal, a second Schmitt trigger input component component proportional to said AC source voltage (βμτT_(AC)/T), and an optional third Schmitt trigger input component proportional to a frequency control signal (V_(FCS)).

FIG. 36 provides an example of ICM-based control of a 3-phase AC/DC converter with high power factor and low harmonic distortions. As one should be able to see in FIG. 36, the switches in the bridge are controlled by three instances of an ICM controller disclosed herein, and each leg of the bridge is separately controlled by its respective ICM controller.

With β given by

$\begin{matrix} {{\beta = {\frac{K^{2}L}{3{\eta\tau}}\frac{I_{load}}{V_{out}}}},} & (15) \end{matrix}$ where η is the converter efficiency, the nominal AC/DC conversion ratio of the 3-phase AC/DC converter shown in FIG. 36 would be

V_(out)

/

V_(LL) ²

^(1/2)=K, where V_(LL) is the nominal line-to-line voltage. Further, additional output voltage regulation based on the difference between the desired nominal (“reference”) and the actual (

V_(out)

) output voltages may be added. For example, a term proportional to said difference may be added to the parameter β.

Also, a desired voltage output V_(ref) may be achieved by using the parameter β that may be expressed as follows:

$\begin{matrix} {\beta = {\frac{3V_{ref}^{2}}{\left\langle V_{ab}^{2} \right\rangle + \left\langle V_{bc}^{2} \right\rangle + \left\langle V_{ca}^{2} \right\rangle}\frac{L}{\eta\tau}{\frac{I_{load}}{V_{out}}.}}} & (16) \end{matrix}$

Note that with the inputs to the integrators of the ICM controllers as shown in FIG. 36, any chosen single node of the power stage (e.g., V₊, V⁻, the “neutral” node V_(N), a switching node a, b, or c, or an AC node V_(a), V_(b), or V_(c)) may be grounded.

We may refer to the voltages V′_(a)=V_(a)−V_(N), V′_(b)=V_(b)−V_(N), and V′_(c)=V_(c)−V_(N) shown in FIG. 36 as line-to-neutral voltages of the 3-phase source voltage. In practical implementations it would be common that the “neutral” node is grounded, i.e., V_(N)=0.

FIG. 37 illustrates the behavior of a particular implementation of a 6 kW ICM-controlled 3-phase AC/DC converter shown in FIG. 36. In this example, the load conductance switches (effectively instantaneously) between 10% and the full load, and the coefficient β is obtained according to equation (16).

Note that the ICM-based control of the converter shown in FIG. 36 would also ensure that the difference between the common mode output voltage V_(CM)=(V₊+V⁻)/2 and the “neutral” node voltage V_(N) would be a zero-mean voltage with the main frequency content at the switching frequencies. This is illustrated in panels (c) and (d) of FIG. 38, which show the PSDs of V_(CM)−V_(N) (with the “neutral” node grounded, i.e., V_(N)=0) for the steady-state operation at full (panel (c)) and 10% (panel (d)) of the nominal load for a particular implementation of a 6 kW ICM-controlled 3-phase AC/DC converter shown in FIG. 36.

Panels (a) and (b) of FIG. 38 show the PSDs of the steady-state inductor currents at full and 10% loads, illustrating low harmonic distortions and the fact that the switching frequency would span a continuous range of values.

FIG. 39 provides an example of ICM-based control of a 3-phase AC/DC converter that is effectively equivalent to that shown in FIG. 36. In this example, an input signal of the integrator in an ICM controller is a linear combination of voltages proportional to (1) a line-to-line voltage (e.g. −μV_(ab)), (2) its time derivative (e.g. βμτ{dot over (V)}_(ab)), and (3) the difference between two respective switching voltages (e.g. μV*_(ab)).

FIG. 40 provides an example of an ICM-controlled 3-phase inverter (DC/AC converter). Here, the output line-to-line voltages V_(ab)=V_(a)−V_(b), V_(bc)=V_(b)−V_(c), and V_(ca)=V_(c)−V_(a) would be proportional to the respective differences between the reference voltages

_(a)−

_(b),

_(b)−

_(c), and

_(c)−

_(a), where

_(a)+

_(b)+

_(c)=0. Note that, in general, the reference voltages do not need to be sinusoidal signals.

We may refer to the voltages V′_(a)=V_(a)−V_(N), V′_(b)=V_(b)−V_(N), and V′_(c)=V_(c)−V_(N) shown in FIG. 40 as line-to-neutral voltages of the 3-phase output voltage. Note that in practical implementations the “neutral” node V_(N) may be floating (not grounded), and thus may be a “virtual neutral”. Whether the node V_(N) is grounded or not, we may refer to it as the “virtual neutral”.

We may also refer to the voltages V_(ab)=V_(a)−V_(b), V_(bc)=V_(b)−V_(c), and V_(ca)=V_(c)−V_(a), and the voltages V′_(a)=V_(a)−V_(N), V′_(b)=V_(b)−V_(N), and V′_(c)=V_(c)−V_(N) shown in FIG. 40, as simply “AC voltages”.

Further note that the virtual neutral is connected to the AC outputs by capacitors, and that the capacitances of these capacitors may or may not be effectively equal.

As one should be able to see in FIG. 40, the switches in the bridge are controlled by three instances of an ICM controller disclosed herein, and each leg of the bridge is separately controlled by its respective ICM controller.

Note that with the inputs to the integrators of the ICM controllers as shown in FIG. 40, the ground may be placed at (connected to) any chosen single node of the power stage (e.g., V₊, V⁻, the “neutral” node V_(N), a switching node a, b, or c, or an AC node V_(a), V_(b), or V_(c)).

Also note that the inputs signals of the integrator in an ICM controller in the inverter shown in FIG. 40 may also be as shown in FIG. 30. In FIG. 30, an input signal of the integrator in an ICM controller is a linear combination of voltages proportional to (1) a difference between two switching voltages (e.g. μV*_(ab)), (2) the time derivative of the respective line-to-line voltage (e.g. μτ{dot over (V)}_(ab)), and (3) the respective reference voltage (e.g.

_(ab)).

FIG. 41 illustrates transient output power, voltages and currents, and the inductor currents, for a particular implementation of a 6 kW ICM-controlled 3-phase inverter shown in FIG. 40, in response to full-range (and independent of each other) step changes in the line-to-line resistive loads.

8 Digital Implementations of ICM Controllers

While the ICM development relies on analog methodology, in many practical deployments low computational cost field-programmable gate array (FPGA) implementations may be a preferred choice, offering on-the-fly control reconfigurability and resistance to on-board electromagnetic interference (EMI). However, to benefit from the analog concept, a sampling rate in a digital ICM implementation would need to be significantly higher (e.g. by two or more orders of magnitude) than the switching frequency.

Since the ICM algorithm does not use products/ratios of the inputs, and/or their nonlinear functions such as coordinate transformations, instead of using high-resolution analog-to-digital converters (ADCs), the desired “effectively analog” controller bandwidth may be easily achieved by using simple 1-bit ΔΣ modulators [8, 9] (e.g., 1st order ΔΣ modulators), as illustrated in FIG. 42.

In the figure, the lowpass filter preceding the numerical Schmitt triggers may be an infinite impulse response (IIR) filter with the bandwidth of order of the typical switching frequency. An IIR filter (e.g. a 2nd order IIR lowpass Bessel filter) may be used in order to both reduce the group delay and lower the computational and memory requirements. Provided that the sampling rate of the ΔΣ modulators is sufficiently high, the performance of such a digital ICM controller may be effectively equivalent to that of the respective analog controller, as illustrated in FIG. 43.

In FIGS. 42 and 43, a numerical integrator may compute a “numerical antiderivative” as follows: “IntOut(i)=IntOut(i−1)+IntIn(i)*dt”, where “dt” is the sampling time interval.

A second order analog lowpass filter may be described by the following differential equation: ζ(t)=z(t)−τ{dot over (ζ)}(t)−(τQ)²{umlaut over (ζ)}(t),  (17) where z(t) and ζ(i) are the input and the output signals, respectively, τ is the time parameter of the filter (inversely proportional to the corner frequency f₀, τ≈1/(2πQf₀)), Q is the quality factor, and the dot and the double dot denote the first and the second time derivatives, respectively. For the Bessel filter, Q=1/√{square root over (3)}.

When the signal sampling rate is sufficiently high (e.g. the sampling interval is much smaller than the time parameter), a finite-difference solution of equation (17) would sufficiently well approximate the analog filter.

An example of such a numerical algorithm for a 2nd order IIR lowpass Bessel filter may be given by the following MATLAB function:

function zeta=Bessel_2nd_order(z,dt,tau)

zeta=zeros(size(z));

zeta(1)=z(1);

T1sq=0.5*dt*tau*sqrt(3);

T2sq=tau^2;

dtsq=dt^;

zeta(2)=(2*T2sq*zeta(1)+zeta(1)*(T1sq−T2sq))/(T2sq+T1sq);

for i=3:length(z);

-   -   dZ=z(i−1)−zeta(i−1);     -   zeta(i)=(dtsq*dZ+2*T2sq*zeta(i−1)+zeta(i−2)*(T1sq−T2sq))/(T2sq+T1sq);

end

return

In this example, “z” is the input signal, “zeta” is the output, “tau” is the time parameter of the filter, and “dt” is the sampling interval.

An example of an algorithm for an inverting numerical Schmitt trigger may be given by the following MATLAB function:

function y=SchmittTriggerInv(x,h0,dh,y_old)

if x<h0, y=1;

elseif x>h0+dh, y=0;

else y=y_old;

end

return

In this example, “x” is the input signal, “y” is the output, “h0” is the lower threshold of the trigger, and “dh” is the hysteresis gap.

One skilled in the art will recognize from the above description that an ICM controller may be implemented in a digital signal processing apparatus performing numerical functions that include numerical integration function, lowpass filtering function, and numerical Schmitt trigger function.

One skilled in the art will also recognize that oversampled modulators with the amplitude resolution higher than 1 bit may be used to obtain digital representations of the analog voltages for digital ICM controllers. For example, the quantizers in such modulators may be realized with N-level comparators, thus the modulators would have log₂(N)-bit outputs. A simple comparator with 2 levels would be a 1-bit quantizer; a 3-level quantizer may be called a “1.5-bit” quantizer; a 4-level quantizer would be a 2-bit quantizer; a 5-level quantizer would be a “2.5-bit” quantizer. A higher quantization level would allow for a wider-bandwidth lowpass filtering.

Further, provided that the sampling rate of the ADCs is sufficiently high, ADCs with even higher amplitude resolution (e.g. 6-bit or higher) may be used to obtain digital representations of the analog voltages for digital ICM controllers. When the ADC amplitude resolution is sufficiently high (e.g. 6-bit or higher), the lowpass filtering function may become optional, and only numerical integration function and numerical Schmitt trigger function would be needed to implement a digital ICM controller. This is illustrated in FIG. 44 for 8-bit digital representations of the analog voltages.

9 Holistic Control of Three-Phase Bidirectional Rectifiers

One skilled in the art will recognize from the description presented in this disclosure that an analog (albeit permitting digital implementation) ICM controller may enable a minimalistic 6-switch bridge to perform as a high-quality multi-level bidirectional three-phase rectifier, holistically incorporating startup, EMI, and overload management, tolerance to mains voltage imbalance and/or phase loss, and offering multiple ways to optimize the cost-size-weight-performance tradespace.

Voltage and current relations in a switching power converter would be best described by a system of continuous-time differential equations, which may be easily solved, in real time, by a simple analog feedback circuit comprising an integrator. Under such a premise, the controller becomes a holistic part of the controller topology rather than a computational add-on. Meeting additional specifications such as effectively unity power factor (PF), low total harmonic distortion (THD), reliable regulation under a wide range of load conditions and/or changes, tolerance to mains voltage imbalance and/or phase loss, startup, electromagnetic interference (EMI), and overload management, operation at variable AC frequencies, bidirectionality, and the ability to connect converters in parallel to increase the output power and/or achieve N+1 redundancy may be innately incorporated into the controller function.

For example, FIG. 45 shows the principal schematic of a 3-phase 6-switch rectifier that incorporates the power stage, EMI filtering (components within the dashed-line boundary), and low-power analog circuitry that provides both the control of the power stage and non-dissipative resonance damping of the EMI filters. As may be seen in FIG. 45, each instance of the ICM controller comprises (1) an integrator (with the time constant T) and (2) two Schmitt triggers (inverting and non-inverting, with the same hysteresis gap Δh) with a slight offset in the reference voltage. Each ICM controller supplies a pair of the switch control signals Q_(ij) to the MOSFET switches in the respective leg of the bridge.

For simplicity, let us first consider the operation of the converter shown in FIG. 45 without the EMI filtering (that is, without the circuit components and the ICM integrator inputs outlined by the dashed-line boundaries). In such a case, with a few sensible idealizations, the ICM controllers would solve a system of the following differential equations:

$\begin{matrix} \left\{ {\begin{matrix} {{\overset{\_}{I}}_{a} = {{\beta\frac{\tau}{L}{\overset{\_}{V}}_{a}^{\prime}} - {\frac{1}{L}{\int{{dt}\left( {{\overset{\_}{V}}_{CM} - {\overset{\_}{V}}_{N}} \right)}}}}} \\ {{\overset{\_}{I}}_{b} = {{\beta\frac{\tau}{L}{\overset{\_}{V}}_{b}^{\prime}} - {\frac{1}{L}{\int{{dt}\left( {{\overset{\_}{V}}_{CM} - {\overset{\_}{V}}_{N}} \right)}}}}} \\ {{\overset{\_}{I}}_{c} = {{\beta\frac{\tau}{L}{\overset{\_}{V}}_{c}^{\prime}} - {\frac{1}{L}{\int{{dt}\left( {{\overset{\_}{V}}_{CM} - {\overset{\_}{V}}_{N}} \right)}}}}} \\ {{I_{a} + I_{b} + I_{c}} = 0} \end{matrix},} \right. & (18) \end{matrix}$ where overlines denote averaging over a time interval between any pair of rising/falling edges of the respective switch control signals Q_(ij). It should be easily seen that, for a balanced system, a steady-state solution of (18) would lead to proportionality between the line voltages and the respective currents, providing 3-phase AC/DC conversion with a nearly unity power factor and low harmonic distortions. Note that the switching behavior of the ICM-based converter shown in FIG. 45 may not be characterized in such simple terms as pulse-width or pulse-frequency modulation (PWM or PFM), and that the voltage at a switching node (e.g. V*_(a)−V_(N)) would exhibit an interleaved 5-level, intrinsically spread-spectrum pattern (see, e.g., FIG. 49) with a frequency range determined by the AC and DC voltages and the values of the parameters μ, T, and Δh (and, when operating in a discontinuous conduction mode at light loads, by the load). Also note that in the configuration shown in FIG. 45 the difference between the common mode output voltage V_(CM)=(V+V⁻)/2 and the “neutral” node voltage V_(N) would be a zero-mean voltage with the main frequency content at the switching frequencies. Further, when the LC EMI filtering network is added, the additional ICM controller inputs (outlined by the dashed-line boundaries, α˜√{square root over (LC)}/τ) would provide non-dissipative resonance damping of the LC circuits in the EMI filters.

We may refer to the voltages V′_(a), V′_(b), and V′_(c) indicated in FIG. 45 as “AC voltages”, or as “AC line voltages”. We may further refer to the voltages {tilde over (V)}′_(a), {tilde over (V)}′_(b), and {tilde over (V)}′_(c) indicated in FIG. 45 as “LC filter voltages”, or “LC damping voltages”.

One skilled in the art will recognize that a variety of LC damping voltages different from those shown in FIG. 45 may be constructed to provide resonance damping of the LC circuits in the EMI filters. For example, the LC damping voltages may be obtained as a combination (e.g. a linear combination) of other node voltages of the LC circuits in the EMI filters.

As follows from equation (18), the steady-state AC power drawn from (or supplied to for β<0) the 3-phase AC source would be

$P_{A\; C} = {\beta\;\frac{\tau}{L}{\left( {\left\langle V_{a}^{\prime\; 2} \right\rangle + \left\langle V_{b}^{\prime\; 2} \right\rangle + \left\langle V_{c}^{\prime\; 2} \right\rangle} \right).}}$ Then a desired regulated voltage output V_(ref) may be achieved by using the parameter β that may be expressed, for example, as

$\begin{matrix} {{\beta = {\frac{V_{ref}^{2}}{\left\langle V_{a}^{\prime 2} \right\rangle + \left\langle V_{b}^{\prime 2} \right\rangle + \left\langle V_{c}^{\prime 2} \right\rangle}\frac{L}{\eta\tau}\frac{I_{load}}{V_{D\; C}}}},} & (19) \end{matrix}$ where η is the converter efficiency, and only the DC-side current measurement may required for such voltage regulation. Equation (19) may provide the basis for voltage regulation of the ICM-based rectifier, and may be modified in various ways to meet the desired specifications (e.g. the right-hand side of (19) may be multiplied by a positive power of V_(ref)/V_(DC) to tighten the voltage regulation). For example, it may be adjusted to adapt to severe mains voltage imbalance and/or phase loss (e.g., by replacing

V′_(a) ²

+

V′_(b) ²

+

V′_(b) ²

with (

V′_(a) ²

+

V′_(b) ²

)/2 for loss of V_(c)), and managing the inrush startup and the overload currents may be achieved my limiting the maximum value of β. Further, changing the sign of β from positive to negative would reverse the power flow, converting a rectifier into an inverter.

FIGS. 46 through 49 provide several examples of behavior and performance of a particular implementation of an ICM-based 3-phase rectifier shown in FIG. 45, for 115 VAC to 400 VDC conversion with ƒ_(AC)=500 Hz. Fully analog ICM controller implementation was used to enable LTspice simulations, and analog behavioral models (ABMs) were used to produce the ICM control signals. High-fidelity LTspice models for the SiC MOSFETs (C2M0025120D) and the SiC Schottky diodes (C4D20120A) were used in the simulations, and, for efficiency/losses breakdown, simplified inductor losses assessment was used, that generally overestimates the magnetics losses as compared with those for carefully designed inductors.

FIG. 46 shows transient responses under mains voltage imbalance, FIG. 47 provides the efficiency/losses breakdown, PF, and THD, FIG. 48 shows the FFTs of the steady-state line currents and CM/DM voltages at 10 kW output, and FIG. 49 shows the steady-state switching voltages at 10 kW output.

For given rectifier components and their values, the power losses in various components would be different nonlinear functions of the load, and would also exhibit different nonlinear dependences on the integrator time constant T and/or the hysteresis gap Δh. Thus, given a particular choice of the switches (e.g. particular MOSFET switches) and their drivers, the magnetics, and other passive rectifier components, by adjusting T and/or Δh one may achieve the best overall compromise among various component power losses (e.g., between the bridge and the inductor losses), while remaining within other constraints on the rectifier specifications. A digital ICM controller implementation would enable efficiency calibration (and/or “on-the-fly” adjustment) of the rectifier by adjusting T and/or Δh based on the output power (or current) to maximize the efficiency for a given load.

9.1 Load/Power Sharing

A plurality of ICM-controlled rectifiers may be connected in parallel, for load/power sharing, e.g. to increase the output power and/or achieve N+1 redundancy.

For example, when N rectifiers are connected to a common AC source, regulation of the voltage output may be achieved by using the parameter β for each ith rectifier that may be expressed, for example, as

$\begin{matrix} {{\beta_{i} = {p_{i}\frac{L_{i}}{\eta_{i}\tau_{i}}\frac{I_{load}}{V_{D\; C}}\frac{V_{ref}^{2}}{\left\langle V_{a}^{\prime 2} \right\rangle + \left\langle V_{b}^{\prime 2} \right\rangle + \left\langle V_{c}^{\prime 2} \right\rangle}}},} & (20) \end{matrix}$ where the index i indicates the quantities for the ith rectifier, and p_(i) is the designated fractional power output of the ith rectifier, Σ_(i) ^(N)p_(i)=1.

Additional ICM controller inputs (e.g. indicative of linear combinations of node voltages) may be used to ensure the desired coordinated performance of load sharing rectifiers, for example, to reduce their startup transients.

10 ICM-Based H-Bridge Converters with 3-Level Switching

The basic “free-running” H-bridge converter (see, e.g., FIGS. 14 and 33) with a single ICM controller is a two-level converter, wherein the switching voltage V* effectively alternates between the two voltage levels ±V_(in). As follows from the detailed description given above, however, ICM controllers may allow us to achieve three-level switching in an H-bridge converter, when the switching voltage V* mostly alternates either between zero and V_(in), or between zero and −V_(in).

Such 3-level switching may be highly desirable as it would improve the converter performance (e.g. reduce THD and EMI) and/or its other specifications. For example, for the same inductor current ripple specifications, we may reduce the size of the inductors and/or reduce the switching frequency. Or, for given inductors, we may reduce both the inductor current ripples and/or the switching frequency.

Three-level switching in an ICM-based H-bridge converter may be achieved by controlling each “leg” of the full H bridge by its own instance of an ICM controller, and providing appropriate control signals to each of the ICM controllers. This is illustrated in FIG. 50 for an H-bridge inverter. By using the inputs to the integrators of the ICM controllers as shown in the figure, the 3-level switching would be enabled.

We may refer to the voltages V′₁=V₁−V_(N) and V′₂=V₂−V_(N) indicated in FIG. 50 as “AC line voltages”. While FIG. 50 shows, as an example, a particular choice of switching and AC line voltages used as controller inputs, one skilled in the art will recognize that different combinations (e.g. linear combinations) of switching and AC line voltages may be used to achieve a desired switching pattern (e.g. a three-level switching pattern).

FIG. 51 illustrates performance of a 3-level ICM-controlled H-bridge inverter in comparison with a respective 2-level inverter with the same power stage components. One may see from the figure that the 3-level control reduces the inductor current ripples, EMI, and THD, while maintaining effectively the same range of switching frequencies.

One skilled in the art will recognize that an LC EMI filtering network may be added to an H-bridge converter in a manner similar to that depicted in FIG. 45 for a 3-phase converter. Further, LC damping voltages may be added as ICM controller inputs to provide resonance damping of the LC circuits in the EMI filters.

11 “Grid Forming” and “Grid Following” Operation of ICM-Based H-Bridge Switching Power Converters

The simulated examples in this section were performed in LTspice, and thus fully analog ICM controller implementation was used to enable the LTspice simulations. Analog Behavioral Models (ABMs) were used to produce the ICM control signals. The 3-level modification of an ICM-based H-bridge converter illustrated in FIG. 52 was used.

As shown in FIG. 52, the two “legs” of the full H bridge are each controlled by its own instance of an ICM controller. In the “bare” implementation, the input to the ith ICM controller may consist of three voltages: (1) the switching voltage V*′_(i) proportional to the voltage at the switching node of the leg (this input would be the same for both the grid forming and the grid following modes of operation), (2) the “AC setting voltage” V_(i,set) (proportional to the grid voltage V′_(i) for the grid following mode, and to the “desired” reference voltage V_(ref) for the grid forming mode), and (3) the “power setting voltage”βV′_(i). The coefficient β would be a positive constant for the grid forming mode, and would be a parameter (time-varying in general) proportional to a desired average “true” power drawn from (for β<0) and/or delivered to (for β>0) the grid while operating in the grid following mode.

An additional 4th (effectively the same for both the grid forming and the grid following modes of operation) ICM control voltage (an LC damping voltage) may be used when common mode (CM) and differential mode (DM) EMI filtering is added to the “bare” converter, to perform non-dissipative resonance damping of the CM and DM LC filters. Such additional EMI filtering would also improve the overall performance of the converters (including, e.g., improving PF at light loads and reducing THD).

In the grid forming mode of operation, an ICM-based H-bridge converter would effectively serve as an AC voltage source, providing an AC output voltage proportional to the “desired” reference voltage V_(ref). In the grid following mode, the converter would either supply (for β>0) an effectively unity power factor (PF) AC power at set level

P_(AC)

∝β (e.g. at optimal power point from a photovoltaic (PV) DC source), or draw (for β<0) AC power at set level

P_(AC)

∝−β (with p.f.≈1) from the grid (e.g. to provide DC current for charging a battery).

Multiple ICM-based H-bridge converters may be directly connected in parallel (e.g., without a need for isolation transformers) to form a microgrid, and the microgrid may further be connected to the main grid.

When connected to the main grid (that would be considered an AC voltage source), all converters in the microgrid would operate in a grid following mode, supplying to and/or drawing power from the main grid at set (and possibly time varying) power levels. In an islanded mode (when disconnected from the main grid), one of the converters would operate in a grid forming mode, while the rest would operate in grid following modes. The designation of the grid forming converter would depend on the available and/or desired supply/demand power levels of each converter (e.g. depending on the AC load and/or charging battery states and/or optimal power points of PV and/or other sources), and may change (e.g. using a hysteretic algorithm) as those power levels change in time.

It may be useful to think of a grid following converter with p.f.≈1 as having an average conductance

G

≈−

P_(AC)

/

V_(AC) ²

(positive when the power is drawn from the grid, and negative when the power is supplied to the grid), to calculate the impedance of the microgrid as seen by the grid forming converter.

One of the main appealing features of the ICM-based converters may be their stability and effectively instantaneous (in comparison with the AC period) synchronization with the grid (for the grid following mode) and/or the reference voltage (for the grid forming mode), e.g., accomplished on the order of tens of microseconds when switching between the two modes.

In the illustrative setup shown in FIG. 53, two paralleled ICM-based H bridge converters (assumed, e.g., to be powered by a 450 VDC battery and a 450 VDC PV DC sources) provide AC power to a highly nonlinear load that consists of a series RL circuit with p.f.=0.5 at 50 Hz (79.4 mH inductor in series with 14.4Ω resistor) in parallel with a full-wave diode rectifier. With no load at the output of the rectifier, the PV-based converter operates in the grid forming mode (at 50 Hz), and the battery-based converter operates in a grid following mode (charging the battery). With a heavy rectifier load, the PV-based converter operates in a grid following mode, providing a set amount of average power (e.g., determined by the optimal power point of the PV source), while the battery-based converter supplements the rest of the power to the load, operating in the grid forming mode (at 60 Hz).

The grid setting voltages V_(ref) of the converters are chosen to produce the same RMS output voltage 240 VAC, but are intentionally not synchronized to each other and have different frequencies (50 Hz and 60 Hz, respectively, for the PV- and battery-powered converter), to illustrate effectively instantaneous synchronization with the microgrid and/or the reference voltage.

With a heavy load, the rectifier dissipates approximately 2 kW, and at no load its dissipation is negligible. At 50 Hz 240 VAC, the RL circuit dissipates approximately 1 kW, and at 60 Hz 240 VAC it dissipates approximately 750 W. Together, at 60 Hz 240 VAC the RL circuit and the rectifier with the heavy load dissipate approximately 2.75 kW.

The optimal power point of the PV source was set at approximately 2 kW, leading to the (negative) conductance of the PV-based converter operating in a grid following mode, as seen by the grid, of approximately −0.0347Ω⁻¹ (−28.8Ω). The charging power of the battery was set at approximately 400 W (≈900 mA charging current), leading to the conductance of the battery-based converter operating in a grid following mode, as seen by the grid, of approximately 0.0069Ω⁻¹ (144Ω).

FIG. 54 shows the DC voltages V_(DC) and V′_(DC), the DC powers (P_(DC)=I_(DC)V_(DC) and P′_(DC)=I′_(DC)V′_(DC)), the output (load) voltage V_(AC) and current I_(load), and the output voltage of the rectifier V_(rec) for the setup of FIG. 53.

As should be easily seen in FIG. 54, the output AC voltage is maintained according to that set by a grid forming converter, while the power sharing between the converters changes depending on the load. Note that the output AC voltage is well maintained even for a highly nonlinear relation between the AC voltage and the load current (i.e. a heavy rectifier load).

With no load at the output of the rectifier, the PV-based converter operates in a grid forming mode, supplying approximately 1 kW to the RL circuit, and approximately 400 W for charging the battery (1.4 kW total). With a heavy rectifier load, the PV-based converter operates in a grid following mode, supplying approximately 2 kW, and the battery-based converter operates in a grid forming mode, supplementing approximately 750 W to the load (2.75 kW total).

12 Grid Forming and Grid Following Operation of ICM-Based 3-Phase Switching Power Converters

Multiple ICM-based 3-phase converters may also be directly connected in parallel (e.g., without a need for isolation transformers) to form a microgrid, and the microgrid may further be connected to the main grid.

The three “legs” of the 6-switch bridge would be each controlled by its own instance of an ICM controller. In the “bare” implementation, the input to an ICM controller would consist of three voltages: (1) the switching voltage V*′_(i) proportional to the voltage at the ith switching node of the leg (this input would be the same for both the grid forming and the grid following modes of operation), (2) the “AC setting voltage” V_(i,set) (proportional to the respective grid voltage V′_(i,AC) for the grid following mode, and to the “desired” reference voltage V_(i,ref) for the grid forming mode), and (3) the “power setting voltage” βV′_(i,AC). The coefficient β would be a positive constant for the grid forming mode, and would be a parameter (time-varying in general) proportional to a desired average “true” power drawn from (for β<0) and/or delivered to (for β>0) the grid while operating in the grid following mode.

An additional 4th (effectively the same for both the grid forming and the grid following modes of operation) ICM control voltage (an LC damping voltage) may be used when common mode (CM) and differential mode (DM) EMI filtering is added to the “bare” converter, to perform non-dissipative resonance damping of the CM and DM LC filters. Such additional EMI filtering would also improve the overall performance of the converters (including improving PF at light loads and reducing THD).

In the grid forming mode of operation, an ICM-based 6-switch 3-phase converter would effectively serve as a 3-phase AC voltage source, providing a 3-phase AC output voltage proportional to the “desired” reference voltage {V_(1,ref), V_(2,ref), V_(3,ref)}. In the grid following mode, the converter would either supply (for β>0) an effectively unity power factor (PF) AC power at set level

P_(AC)

∝β (e.g. at optimal power point from a photovoltaic (PV) DC source), or draw (for β<0) AC power at set level

P_(AC)

∝−β (with p.f.≈1) from the grid (e.g. to provide DC current for charging a battery).

When connected to the main grid (that would be considered a 3-phase AC voltage source), all converters in the microgrid would operate in a grid following mode, supplying to and/or drawing power from the main grid at set (and possibly time varying) power levels. In an islanded mode (when disconnected from the main grid), one of the converters would operate in a grid forming mode, while the rest would operate in grid following modes. The designation of the grid forming converter would depend on the available and/or desired supply/demand power levels of each converter (e.g. depending on the AC load and/or charging battery states and/or optimal power points of PV and/or other sources), and may change (e.g. using a hysteretic algorithm) as those power levels change in time.

REFERENCES

-   [1] Qing-Chang Zhong and T. Hornik. Control of Power Inverters in     Renewable Energy and Smart Grid Integration. Wiley, 2013. -   [2] R. Bracewell. The Fourier Transform and Its Applications,     chapter “Heaviside's Unit Step Function, H(x)”, pages 61-65.     McGraw-Hill, New York, 3rd edition, 2000. -   [3] P. A. M. Dirac. The Principles of Quantum Mechanics. Oxford     University Press, London, 4th edition, 1958. -   [4] A. V. Nikitin, “Method and apparatus for control of     switched-mode power supplies.” U.S. Pat. No. 9,130,455 (8 Oct.     2015). -   [5] A. V. Nikitin, “Switched-mode power supply controller.” U.S.     Pat. No. 9,467,046 (11 Oct. 2016). -   [6] “Little box challenge,” 26 Mar. 2016. [Online]. Available:     https://en.wikipedia.org/wiki/Little_Box_Challenge -   [7] “Detailed inverter specifications, testing procedure, and     technical approach and testing application requirements for the     little box challenge,” 16 Jul. 2015. [Online]. Available:     https://www.littleboxchallenge.com/pdf/LBC-InverterRequirements-20150717.pdf -   [8] G. I. Bourdopoulos, A. Pnevmatikakis, V. Anastassopoulos,     and T. L. Deliyannis. Delta-Sigma Modulators: Modeling. Design and     Applications. Imperial College Press, London, 2003. -   [9] Y. Geerts, M. Steyaert, and W. M. C. Sansen. Design of Multi-Bit     Delta-Sigma A/D Converters. The Springer International Series in     Engineering and Computer Science. Springer US, 2006.

Regarding the invention being thus described, it will be obvious that the same may be varied in many ways. Such variations are not to be regarded as a departure from the spirit and scope of the invention, and all such modifications as would be obvious to one skilled in the art are intended to be included within the scope of the claims. It is to be understood that while certain now preferred forms of this invention have been illustrated and described, it is not limited thereto except insofar as such limitations are included in the following claims. 

I claim:
 1. A switching converter capable of converting a 3-phase AC source voltage into a DC output voltage and providing a DC power output, wherein said 3-phase AC source voltage is characterized by three AC voltages, wherein an AC voltage is one of said three AC voltages, wherein said DC output voltage is characterized by a DC common mode voltage, wherein said switching converter comprises a 3-phase bridge comprising three pairs of switches and capable of providing three switching voltages, wherein a switching voltage is provided by a pair of switches controlled by a controller providing a 1st control signal and a 2nd control signal, and wherein a 1st switch of said pair of switches is controlled by said 1st control signal and a 2nd switch of said pair of switches is controlled by said 2nd control signal, said switching converter further comprising a digital signal processing apparatus performing numerical functions including: a) a numerical integrator function operable to receive an integrator input and to produce an integrator output, wherein said integrator output is proportional to a numerical antiderivative of said integrator input; b) a digital lowpass filter operable to receive a filter input and to produce a filter output; c) a 1st numerical Schmitt trigger characterized by a hysteresis gap and a 1st reference level, and operable to receive a Schmitt trigger input and to produce said 1st control signal; and d) a 2nd numerical Schmitt trigger characterized by said hysteresis gap and a 2nd reference level, and operable to receive said Schmitt trigger input and to produce said 2nd control signal; wherein said integrator input comprises a sum of a 1-bit digital representation of said AC voltage and a 1-bit digital representation of a difference between said switching voltage and said DC common mode voltage, wherein said filter input comprises said integrator output, and wherein said Schmitt trigger input comprises said filter output.
 2. The switching converter of claim 1 wherein said DC output voltage is characterized by a desired DC voltage value, wherein said filter input further comprises a component proportional to said 1-bit digital representation of said AC voltage, and wherein a magnitude of said component proportional to said 1-bit digital representation of said AC voltage is chosen to provide said desired DC voltage value.
 3. The switching converter of claim 1 wherein said DC power output is characterized by a desired DC power value, wherein said filter input further comprises a component proportional to said 1-bit digital representation of said AC voltage, and wherein a magnitude of said component proportional to said 1-bit digital representation of said AC voltage is chosen to provide said desired DC power value.
 4. The switching converter of claim 1 further comprising an LC EMI filtering network, wherein said filter input further comprises a component proportional to a 1-bit digital representation of an LC damping voltage, and wherein said LC damping voltage is chosen to provide resonance damping of said LC EMI filtering network.
 5. A switching converter capable of converting a DC source voltage into a 3-phase AC output voltage and providing an AC power output, wherein said 3-phase AC output voltage is characterized by three AC voltages indicative of respective AC setting voltages, wherein an AC voltage is one of said three AC voltages, wherein said DC source voltage is characterized by a DC common mode voltage, wherein said switching converter comprises a 3-phase bridge comprising three pairs of switches and capable of providing three switching voltages, wherein a switching voltage is provided by a pair of switches controlled by a controller providing a 1st control signal and a 2nd control signal, and wherein a 1st switch of said pair of switches is controlled by said 1st control signal and a 2nd switch of said pair of switches is controlled by said 2nd control signal, said switching converter further comprising a digital signal processing apparatus performing numerical functions including: a) a numerical integrator function operable to receive an integrator input and to produce an integrator output, wherein said integrator output is proportional to a numerical antiderivative of said integrator input; b) a digital lowpass filter operable to receive a filter input and to produce a filter output; c) a 1st numerical Schmitt trigger characterized by a hysteresis gap and a 1st reference level, and operable to receive a Schmitt trigger input and to produce said 1st control signal; and d) a 2nd numerical Schmitt trigger characterized by said hysteresis gap and a 2nd reference level, and operable to receive said Schmitt trigger input and to produce said 2nd control signal; wherein said integrator input comprises a sum of a 1-bit digital representation of said AC setting voltage and a 1-bit digital representation of a difference between said switching voltage and said DC common mode voltage, wherein said filter input comprises said integrator output, and wherein said Schmitt trigger input comprises said filter output.
 6. The switching converter of claim 5 wherein said AC voltage is characterized by a desired AC voltage and wherein said AC setting voltage is proportional to said desired AC voltage.
 7. The switching converter of claim 5 wherein said AC power output is characterized by a desired AC power output, wherein said AC setting voltage is proportional to said AC voltage, wherein said filter input further comprises a component proportional to a 1-bit digital representation of said AC voltage, and wherein a magnitude of said component proportional to said 1-bit digital representation of said AC voltage is chosen to provide said desired AC power output.
 8. The switching converter of claim 5 further comprising an LC EMI filtering network, wherein said filter input further comprises a component proportional to a 1-bit digital representation of an LC damping voltage, and wherein said LC damping voltage is chosen to provide resonance damping of said LC EMI filtering network.
 9. An AC/DC converter capable of converting an AC source voltage into a DC output voltage and providing a DC power output, wherein said DC output voltage is characterized by a DC common mode voltage, wherein said AC source voltage is characterized by two AC line voltages, wherein an AC voltage is one of said two AC line voltages, wherein said AC/DC converter comprises an H bridge comprising two pairs of switches and capable of providing two switching voltages, wherein a switching voltage is provided by a pair of switches controlled by a controller providing a 1st control signal and a 2nd control signal, and wherein a 1st switch of said pair of switches is controlled by said 1st control signal and a 2nd switch of said pair of switches is controlled by said 2nd control signal, said switching converter further comprising a digital signal processing apparatus performing numerical functions including: a) a numerical integrator function operable to receive an integrator input and to produce an integrator output, wherein said integrator output is proportional to a numerical antiderivative of said integrator input; b) a digital lowpass filter operable to receive a filter input and to produce a filter output; c) a 1st numerical Schmitt trigger characterized by a hysteresis gap and a 1st reference level, and operable to receive a Schmitt trigger input and to produce said 1st control signal; and d) a 2nd numerical Schmitt trigger characterized by said hysteresis gap and a 2nd reference level, and operable to receive said Schmitt trigger input and to produce said 2nd control signal; wherein said integrator input comprises a sum of a 1-bit digital representation of said AC voltage and a 1-bit digital representation of a difference between said switching voltage and said DC common mode voltage, wherein said filter input comprises said integrator output, and wherein said Schmitt trigger input comprises said filter output.
 10. The AC/DC converter of claim 9 wherein said DC output voltage is characterized by a desired DC voltage value, wherein said filter input further comprises a component proportional to said 1-bit digital representation of said AC voltage, and wherein a magnitude of said component proportional to said 1-bit digital representation of said AC voltage is chosen to provide said desired DC voltage value.
 11. The AC/DC converter of claim 9 wherein said DC power output is characterized by a desired DC power value, wherein said filter input further comprises a component proportional to said 1-bit digital representation of said AC voltage, and wherein a magnitude of said component proportional to said 1-bit digital representation of said AC voltage is chosen to provide said desired DC power value.
 12. The AC/DC converter of claim 9 further comprising an LC EMI filtering network, wherein said filter input further comprises a component proportional to a 1-bit digital representation of an LC damping voltage, and wherein said LC damping voltage is chosen to provide resonance damping of said LC EMI filtering network.
 13. A DC/AC converter capable of converting a DC source voltage into an AC output voltage and providing an AC power output, wherein said AC output voltage is characterized by two AC line voltages indicative of respective AC setting voltages, wherein an AC voltage is one of said two AC voltages, wherein said DC source voltage is characterized by a DC common mode voltage, wherein said DC/AC converter comprises an H bridge comprising two pairs of switches and capable of providing two switching voltages, wherein a switching voltage is provided by a pair of switches controlled by a controller providing a 1st control signal and a 2nd control signal, and wherein a 1st switch of said pair of switches is controlled by said 1st control signal and a 2nd switch of said pair of switches is controlled by said 2nd control signal, said switching converter further comprising a digital signal processing apparatus performing numerical functions including: a) a numerical integrator function operable to receive an integrator input and to produce an integrator output, wherein said integrator output is proportional to a numerical antiderivative of said integrator input; b) a digital lowpass filter operable to receive a filter input and to produce a filter output; c) a 1st numerical Schmitt trigger characterized by a hysteresis gap and a 1st reference level, and operable to receive a Schmitt trigger input and to produce said 1st control signal; and d) a 2nd numerical Schmitt trigger characterized by said hysteresis gap and a 2nd reference level, and operable to receive said Schmitt trigger input and to produce said 2nd control signal; wherein said integrator input comprises a sum of a 1-bit digital representation of said AC setting voltage and a 1-bit digital representation of a difference between said switching voltage and said DC common mode voltage, wherein said filter input comprises said integrator output, and wherein said Schmitt trigger input comprises said filter output.
 14. The DC/AC converter of claim 13 wherein said AC voltage is characterized by a desired AC voltage and wherein said AC setting voltage is proportional to said desired AC voltage.
 15. The DC/AC converter of claim 13 wherein said AC power output is characterized by a desired AC power output, wherein said AC setting voltage is proportional to said AC voltage, wherein said filter input further comprises a component proportional to a 1-bit digital representation of said AC voltage, and wherein a magnitude of said component proportional to said 1-bit digital representation of said AC voltage is chosen to provide said desired AC power output.
 16. The DC/AC converter of claim 13 further comprising an LC EMI filtering network, wherein said filter input further comprises a component proportional to a 1-bit digital representation of an LC damping voltage, and wherein said LC damping voltage is chosen to provide resonance damping of said LC EMI filtering network. 